Thread: Prove the following are equivalent

Let R be a nonzero ring. Prove that the following are equivalent:

a)R is a field
b)The only ideals in R are (0) and (1).
c)Every homomorphism of R into a nonzero ring S is injective.

Originally Posted by apple2010

Let R be a nonzero ring. Prove that the following are equivalent:

a) R is a field
b) The only ideals in R are (0) and (1).
c) Every homomorphism of R into a nonzero ring S is injective.

you forgot to mention that R is commutative.

a) --> b) is trivial by the definition of a field. for b) --> c) look at the kernel of the homomorphism, which is an ideal of R.

for c) --> a) suppose that r is a non-zero element of R and put I = Rr, S = R/I and take f : R ---> S to be the natural homomorphism.